Since the tunnel is in the shape of a semi-ellipse, we can use the formula for the equation of a semi-ellipse:
(x^2 / a^2) + (y^2 / b^2) = 1
where "a" is the horizontal radius (half of the width) and "b" is the vertical radius (half of the height).
In this case, we have:
a = 20/2 = 10 feet
b = 12/2 = 6 feet
We can assume that the train is centered in the tunnel, so we need to find the height of the semi-ellipse at the center (i.e., the value of "y" when "x" is 0).
Plugging in the values for "a" and "b", we get:
(0^2 / 10^2) + (y^2 / 6^2) = 1
Simplifying, we get:
y^2 / 36 = 1
y^2 = 36
y = ±6 feet
Therefore, the height of the semi-ellipse at the center is 6 feet.
To determine whether a 10-foot high train would have clearance to pass through, we need to check whether the height of the semi-ellipse at the sides is greater than or equal to 10 feet.
Plugging in the values for "a" and "b" and solving for "y" when "x" is 5 feet (half of the train's width), we get:
(5^2 / 10^2) + (y^2 / 6^2) = 1
Simplifying, we get:
y^2 / 36 = 0.75
y^2 = 27
y ≈ ±5.2 feet
Since the height of the semi-ellipse at the sides is about 5.2 feet, a 10-foot high train would not have clearance to pass through. The clearance is less than 5.2 - 5 = 0.2 feet (or about 2.4 inches).
Therefore, the train would not fit through the tunnel with 10 feet of clearance.